Circles and angle measure

Learning objectives

Distinguish radii, diameters, chords, arcs, and tangents; connect central and inscribed angles; convert between degrees and radians; compute arc lengths and sector areas; use signed unit-circle coordinates and recognize coterminal angles.

Circle vocabulary and the geometry it unlocks

A circle is the set of points in a plane at a fixed distance r from a center. A radius joins the center to the circle. A diameter is a chord through the center and has length 2r. A chord joins two points on the circle. An arc is part of the circle itself; its length is measured along the curve, not across a chord.

A tangent touches a circle at one point. The radius to the point of tangency is perpendicular to the tangent. A secant intersects the circle at two points. Draw the radius to a tangent point whenever you need a right triangle; it often supplies the hidden 90° angle.

The circumference and area of the disk bounded by the circle are

$C = 2πr = πd,$ $A = πr 2 .$

Circumference is a length; area is measured in square units. Doubling a radius doubles circumference but quadruples area.

Angles on and inside a circle

A central angle has its vertex at the center. Its degree measure equals the degree measure of its intercepted arc. An inscribed angle has its vertex on the circle and its sides along chords. Its measure is half the measure of the intercepted arc not containing the vertex.

Consequently, inscribed angles intercepting the same arc are equal. An inscribed angle intercepting a semicircle is 90°, so a triangle with a diameter as one side and its third vertex on the circle is a right triangle. In a quadrilateral with all four vertices on a circle, opposite angles are supplementary: their intercepted arcs together fill the circle.

Left: central and inscribed angles intercept the same arc. Right: the tangent is perpendicular to the radius at T.

Chord and tangent consequences

A perpendicular from the center to a chord bisects the chord. Thus a radius, the center-to-chord distance, and half the chord form a right triangle. Equal chords in the same circle have equal distances from the center and equal minor arcs.

Two tangent segments from the same external point have equal lengths: the two right triangles share the center-to-external-point hypotenuse and have equal radii. The angle between a tangent and a chord is half the intercepted arc measure. For two tangents from one external point, the angle between the tangent segments equals 180° minus the smaller central angle between the radii to the tangent points. The latter follows from the angle sum of a quadrilateral with two right angles.

Arc lengths, sectors, and radians

A sector is a region bounded by two radii and their connecting arc. It is not the same as a circular segment, which is bounded by a chord and an arc. For a central angle of θ°, the fraction of a full circle is $θ/360.$ Use that same fraction of circumference or area: $\frac{θ}{360}$ $\frac{θ}{360}$

A sector’s perimeter includes both radii as well as the arc: $P = 2r + s$ for an ordinary sector with two radial boundary edges. Radians measure angle by an arc-to-radius ratio. If an arc of length s subtends an angle θ at the center of a circle of radius r, then $θ = s/r$ radians. A full circumference gives $2πr/r = 2π$ radians. Therefore

$180° = π radians,$ $\frac{π}{180}$ $\frac{180}{π}$

When θ is in radians, the formulas simplify to

$s = rθ,$ $\frac{1}{2}$

The identity $A sector = rs/2$ is especially useful when a problem gives area and arc length but hides the angle. For an ordinary nonoverlapping sector, use its specified central sweep, between 0 and 2π.

Three objects, three measurements

An arc can have a degree or radian measure (an angle) and an arc length (a distance). These are not interchangeable. A 60° arc has angle measure 60° in every circle, but its length grows with the radius. A chord joining the arc’s endpoints is shorter than the nonzero minor arc and is not computed by the arc-length formula.

The unit circle: signs, exact values, and rotations

On the circle of radius 1 centered at the origin, start at (1, 0) and rotate counterclockwise through angle θ. The terminal point is $(x, y) = (cos θ, sin θ),$ $x 2 + y 2 = 1.$

When $x \neq 0,$ $tan θ = y/x.$ This extends the right-triangle ratios to angles outside the first quadrant. A reference angle is the acute angle between the terminal ray and the horizontal axis. Use its familiar 30°, 45°, or 60° magnitudes, then assign signs from the quadrant.

Quadrant	Cosine (x)	Sine (y)	Tangent (y/x)
I: upper right	Positive	Positive	Positive
II: upper left	Negative	Positive	Negative
III: lower left	Negative	Negative	Positive
IV: lower right	Positive	Negative	Negative

At 0, π/2, π, and 3π/2, the unit-circle points are (1, 0), (0, 1), (−1, 0), and (0, −1), respectively. Tangent is undefined where $x = 0.$

In quadrant II, the horizontal coordinate is negative and the vertical coordinate is positive.

Angles that differ by a whole number of full rotations are coterminal: $θ + 2πk$ has the same terminal point as θ for any integer k. Subtract or add multiples of 2π to reduce an angle to a familiar interval. Negative angles represent clockwise rotation. This reduction preserves sine and cosine values, but do not confuse a multi-turn travel distance with the arc of a single sector.

15 worked examples

4.01. Circumference first, area second

A circle has circumference 18π. What is its area?

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Recognize the structure. Circumference determines the radius; area uses the radius squared.

Work it through. From $2πr = 18π,$ divide by 2π to get $r = 9.$ Then

$A = πr 2 = π(9) 2 = 81π.$

Answer: 81π square units.

Avoid the trap. The value 18 is the diameter, not the radius. Substituting 18 as the radius would multiply the correct area by 4.

4.02. Arc length as a fraction of circumference

A circle has radius 10. A central angle of 72° intercepts a minor arc. Find the arc’s length.

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Recognize the structure. The arc occupies $72/360 = 1/5$ of a full circle.

Work it through. The circumference is $2π(10) = 20π.$ Therefore

$\frac{72}{360}$

Answer: 4π units.

Key takeaway. An arc-length answer has length units. Using $πr 2$ here would compute a sector area rather than a distance around the circle.

4.03. Sector area as a fraction of the disk

A sector has radius 8 and central angle 135°. What is its area?

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Recognize the structure. Use the fraction of the full disk, not the circumference.

Work it through. The full disk has area 64π. The sector occupies $135/360 = 3/8$ of it, so

$\frac{3}{8}$

Answer: 24π square units.

Avoid the trap. The angle is less than 180°, so the area should be less than half the disk’s area, 32π. This offers a quick magnitude check.

4.04. Recover an angle from an arc length

An arc of a circle of radius 12 has length 9π. What is the measure, in degrees, of its central angle?

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Recognize the structure. Find the fraction of the circumference represented by the arc.

Work it through. The circumference is 24π, so the arc is $9π/(24π) = 3/8$ of a full turn. The angle is

$\frac{3}{8}$

Answer: 135°.

Another efficient route. Use $θ = s/r = 9π/12 = 3π/4$ radians, then convert to degrees. The two routes should agree.

4.05. Degrees to radians

What is 225° in radians? A) 4π/5 B) 5π/4 C) 5π/2 D) 225π

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Recognize the structure. A 180° angle corresponds to π radians.

Work it through. Multiply by π/180: $\frac{π}{180}$

Answer: B, 5π/4 radians.

Key takeaway. 225° lies between 180° and 270°, so its radian measure must lie between π and 3π/2. The answer does.

4.06. Use radians without a conversion detour

A sector has radius 9 and central angle 2π/3 radians. Find its area.

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Recognize the structure. The angle is already in radians, so use $\frac{1}{2}$

Work it through. Substitute directly:

$\frac{1}{2}$

Answer: 27π square units.

Another efficient route. The angle is one-third of 2π, so the sector is one-third of the full disk area 81π.

Avoid the trap. Do not put $2π/3$ into the degree formula’s $θ/360$ fraction. The denominator must match the angle unit.

4.07. A sector’s perimeter includes straight edges

A quarter-circle sector has radius 6. What is its perimeter? A) 3π B) 12 + 3π C) 6 + 3π D) 12 + 9π

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Recognize the structure. Trace the entire boundary: two radii and one quarter-circumference arc.

Work it through. The curved portion has length $(1/4)(2π \cdot 6) = 3π.$ Add the two straight radial edges:

$P = 6 + 6 + 3π = 12 + 3π.$

Answer: B, $12 + 3π$ units.

Avoid the trap. Choice A is only the arc length. Choice D mixes the correct straight boundary with the sector’s area, 9π.

4.08. Inscribed angles measure half their arcs

Points A, B, and C lie on a circle with center O. Inscribed angle ∠ACB measures 37°. What is the measure of the central angle subtending the same arc AB not containing C?

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Recognize the structure. A central angle and an inscribed angle intercept the same arc; the central angle is twice as large.

Work it through. The intercepted arc has measure $2(37°) = 74°.$ Its central angle also measures 74°.

Answer: 74°.

Avoid the trap. Use the specified arc not containing C. The other arc has measure $360 - 74 = 286°$ and is not the one intercepted by ∠ACB.

4.09. A diameter supplies the missing right angle

AB is a diameter of a circle, AB = 10, and C is a different point on the circle. If AC = 6, find BC.

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Recognize the structure. An angle inscribed in a semicircle is right, so $∠ACB = 90°.$

Work it through. In right triangle ACB, the diameter AB is the hypotenuse. Hence

$62 + BC 2 = 10 2 BC 2 = 64 BC = 8.$

Answer: 8 units.

Key takeaway. The right angle follows from the diameter condition. Three arbitrary points on a circle do not necessarily form a right triangle.

4.10. A tangent creates a right triangle

From an external point P, segment PT is tangent to a circle at T. The circle’s center is O, its radius is 7, and OP = 25. Find PT.

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Recognize the structure. Draw OT; a radius is perpendicular to the tangent at the contact point.

Work it through. Triangle OTP is right at T, with hypotenuse $OP = 25.$ Thus

$\sqrt{25^{2} − 7^{2}}$

Answer: 24 units.

Avoid the trap. The radius and tangent segment are legs. The center-to-external-point segment, not the tangent, is the hypotenuse.

4.11. The center-to-chord perpendicular bisects the chord

A circle has radius 13. A chord lies at a perpendicular distance of 5 from the center. What is the chord’s length?

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Recognize the structure. The perpendicular from the center meets the chord at its midpoint.

Work it through. Let half the chord be a. A radius to either endpoint gives a right triangle with legs 5 and a, so $a 2 + 5 2 = 13 2 a 2 = 144 a = 12.$

The whole chord has length $2a = 24.$

Answer: 24 units.

Avoid the trap. 12 is half the chord. Also, $13 - 5 = 8$ is a radial gap, not a chord length.

4.12. Opposite angles in a cyclic quadrilateral

All four vertices of quadrilateral ABCD lie on one circle, in that order. If ∠A = (3x + 15)° and ∠C = (5x + 5)°, find ∠A.

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Recognize the structure. Opposite inscribed angles intercept arcs whose total measure is 360°, so the angles total 180°.

Work it through. Solve

$(3x + 15) + (5x + 5) = 180 8x + 20 = 180 x = 20.$

Thus $∠A = 3(20) + 15 = 75°;$ the opposite angle is 105°.

Answer: 75°.

Avoid the trap. The supplementary relationship belongs to opposite angles of a cyclic quadrilateral. Adjacent angles need not be supplementary.

4.13. A signed unit-circle coordinate

A point on the unit circle lies in quadrant II and has x-coordinate −3/5. If its position corresponds to angle θ, what is tan θ?

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Recognize the structure. The coordinates are (cos θ, sin θ), and the quadrant determines the sign of the unknown coordinate.

Work it through. From $x 2 + y 2 = 1,$ $\frac{9}{25}$

Quadrant II has positive y, so $y = 4/5.$ Hence

$\frac{y}{x}$

Answer: −4/3.

Avoid the trap. The square-root step produces two algebraic signs. The quadrant resolves which one is geometrically correct.

4.14. Reduce a rotation, then use reference-angle signs

If θ = 19π/6, what is the exact value of sin θ + cos θ?

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Recognize the structure. Subtract a full turn to locate the terminal point.

Work it through. Since $2π = 12π/6,$ angle 19π/6 is coterminal with 7π/6. This angle is in quadrant III with reference angle $π/6 = 30°.$ Both coordinates are negative:

$\frac{1}{2}$ $\frac{\sqrt{3}}{2}$

Their sum is $\sqrt{3}$

Answer: $\frac{1 + \sqrt{3}}{2}$

Key takeaway. A full rotation changes neither coordinate. The quadrant determines the signs; the reference angle determines the magnitudes.

4.15. Eliminate the unknown central angle

A sector has area 30π square units and arc length 5π units. What is its radius?

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Recognize the structure. Area and arc length share the same angle factor; eliminate it rather than introducing another unknown to solve separately.

Work it through. Using radians, $s = rθ$ and $\frac{1}{2}$ Substitute $rθ = s$ to obtain $A = rs/2.$ Therefore

$\frac{2A}{s}$

Answer: 12 units.

Key takeaway. The implied angle is $s/r = 5π/12 = 75°,$ a valid sector angle. Its perimeter would be $24 + 5π,$ not 5π.

Mastery check and error prevention

Draw the center and radii when a circle problem contains tangents, chords, or inscribed triangles. State whether an angle is central or inscribed before using its arc. State the angle unit before using an arc or sector formula. For unit-circle questions, determine the quadrant before choosing a square-root sign.

Optional recognition tool: chord and secant products Some circle configurations can be reduced by similarity to product relationships. If chords AB and CD intersect at an interior point X, then $XA \cdot XB = XC \cdot XD.$ If two secants from an exterior point P meet the circle in the orders P, A, B and P, C, D, then $PA \cdot PB = PC \cdot PD.$ A tangent PT gives $PT 2 = PA \cdot PB.$ On a secant, multiply the external segment by the whole distance to the far intersection, not by the inside segment alone. Treat these as additional recognition tools; the main lessons above develop the circle relationships used throughout this guide.